In the Pink model, the lipid bilayer is assumed to consist of two independent monolayers.
Each monolayer is represented by a triangular 2D lattice consisting ofN sites, and each lattice site contains a single lipid chain. Each lipid molecule comprises two independent hydrophobic acyl chains and a hydrophilic polar head. The polar heads are transla-tionally frozen to the lattice, and no particular structure for the polar head groups is assumed. The only degrees of freedom included in the Pink model are the acyl chain conformations. These are not simulated directly (i.e., one does not explicitly model the carbon atoms) but are captured in a coarse-grained fashion whereby the chain conforma-tions are grouped intoα= 1, . . . , qdiscrete states. The original Pink model usesq = 10, but we will consider different values also. These states include the ground state (α = 1),
State α Eα lα Dα
Table 3.1: The coarse-graining parameters used to describe the acyl chain conformations in the q = 10 Pink model [Pink et al., 1980a,b; Caill´e et al., 1980;Mouritsen et al.,1983]. For each state conformationα, we list the internal energy Eα, the projected length lα, and the degeneracy Dα. The energy of a single gauche bond equals Γ = 0.45×10−13erg, while M denotes the number of carbon atoms in the chain.
eight low-energy excitations (α = 2, . . . , q−1), while all remaining conformations are grouped into a single disordered state (α = q). Each state α is characterized by three coarse-graining parameters, namely an internal energy Eα, a cross-sectional area Aα, and a degeneracy Dα counting the number of chain conformations with energy Eα and area Aα.
3.2.1 Coarse-graining parameters
To determine the coarse-graining parameters, we assume that a single acyl chain consists ofi= 1, . . . , M carbon atoms, thereby containingM−1 carbon-carbon bonds, and that bonds are either in atrans or gauche configuration. Thetrans configuration yields the lowest energy, while the gauche configuration has a slightly higher energy. The energy difference between the trans and gauche configuration is denoted Γ (Table 3.1). To understand the difference in geometry betweentransandgauchebonds consider a chain segment of four consecutive carbon atoms. The positions of the first three atoms define a two-dimensional plane. In the trans configuration, the fourth atom remains in the plane, while in the gauche configuration, it leaves the plane, and it can do so inward or outward. Thus, each gauche bond is twofold degenerate. In the Pink model, it is assumed that each 2nthgauche bond takes the chain back to the original plane, and so
thegauche degeneracy is given by
G= 2ceil(n/2) , (3.1)
wherendenotes the total number ofgauchebonds in the chain, and where the function ceil means “rounding up” to the nearest integer.
It is convenient to mathematically represent the chain conformations on a hexagonal lattice with next-nearest neighbor distance 2a. We emphasize that this lattice is merely an aid to identify the low-energy chain conformations which are needed to set the coarse-graining parameters: It should not be confused with the triangular simulation lattice on which the Pink Hamiltonian will eventually be defined. The carbon atoms are placed on the nodes of the hexagonal lattice following certain rules, and nearest-neighbor con-nections between atoms represent carbon-carbon bonds. The ground stateα= 1 corre-sponds to the chain conformation that is maximally stretched (Fig.3.1A). Note that, in the ground state, the atoms are alternatingly placed on the left and right lattice node, yielding a characteristic “zig-zag” pattern. The ground state by definition contains only transbonds, its internal energy is set to zero as a reference E1 = 0, and it is obviously nondegenerateD1= 1. The cross-sectional area of the ground state has experimentally been determined as A1 = 20.4 ˚A2 [Pink et al.,1980b]. We also introduce the projected lengthl of the conformation, defined as the difference in thez coordinate between the carbon atom closest to the head group (i = 1) and the one furthest away (i = M), with the z direction as indicated in the figure. For the ground state, it follows that l1 = (M−1)a.
The eight low-energy excitations (α= 2, . . . ,9) are obtained by systematically incorpo-rating gauche bonds. The effect of such a bond is to disrupt the “zig-zag” pattern of the ground state; that is, one no longer places the atoms alternatingly on left and right nodes, but also allows for “excursions” whereby for two consecutive atoms the same direction is chosen. Each such excursion corresponds to a gauchebond, and has energy cost Γ. The gauche bonds are introduced according to the following rules: (1) The two bonds in the chain closest to the head group must always be in the transconfiguration.
In Fig. 3.1, these correspond to the bonds between atoms 1 and 2, and 2 and 3. (2) At most threegauche bonds are allowed, and each time such a bond is included there is an energy cost Γ. (3) The projected chain length l must obey l1 −l ≤ 3a. (4) The acyl chain cannot fold back onto itself. In the coordinate system of Fig.3.1, this means that thez coordinates of the atoms must obey zi+1≥zi.
5
Figure 3.1: Typical chain conformations of the Pink model with M = 7, showing (numbered) carbon atoms placed on the nodes of a hexagonal lattice.
The atom connected to the head group is labeledi= 1 but for clarity the head group is only drawn for the ground state. The zdirection indicates the bilayer normal, while the vertical double arrows indicate the projected length. (A) The ground stateα= 1, consisting of onlytransbonds. (B) The two conformations that constitute the first excited stateα= 2 containing onegauchebond (marked with a cross). (C) Conformation belonging to the second excited state α = 3.
The internal energy is the same as in (B) but the projected length is shorter (the otherα= 3 conformation has the gauche bond between atoms 3 and 4).
Following these rules, we show in Fig. 3.1B the chain conformations (i) and (ii) that form the first excited state α = 2. In (i), a single gauche bond is placed at the very chain end, while in (ii) it is placed at the second-last position. One immediately sees that both conformations have the same energy E2 = Γ, and the same projected length l2 = (M−2)a. To compute the cross-sectional area, one assumes volume conservation for the lipid chains: Aαlα = A1l1. Hence, from the (known) ground state values, the cross-sectional area of the excited state follows. Note that, by placing thegauche bond at the third-last position (Fig. 3.1C), a shorter projected length is obtained, and so conformation (C) does not belong to the first excited state (even though it has the same energy). The total degeneracy of the first exited stateD2= 4, which is the total number of conformations, multiplied by thegauchedegeneracy of Eq. (3.1). The coarse-graining parameters of the remaining excited states can be found analogously, and are listed for completeness in Table 3.1. Finally, for the completely disordered stateα=q= 10, one assumesE10= (0.42M−3.94)×10−13erg,A10= 34 ˚A2, and degeneracyD10= 6×3M−6, which have their origins in experimental considerations [Caill´e et al.,1980].
3.2.2 Pink model Hamiltonian
Having specified the coarse-graining parameters, the Hamiltonian of the Pink model can be written as [Mouritsen,1984]
HPink=H0+HVDW+HP . (3.2)
The first term is the total internal energy of the acyl chains H0 =PN
i=1Es(i), with the sum over all N sites of the triangular lattice, and s(i) ∈ {1, . . . , q} the conformational state at theith lattice site. The second term represents the anisotropic van der Waals interaction between adjacent acyl chains HVDW =−J0P
hi,jiIs(i)Is(j), with J0 the van der Waals coupling constant, and hi, ji a sum over all 3N nearest-neighboring sites on the triangular lattice. The precise value ofJ0 depends on the chain length, and explicit expressions are provided elsewhere [Pink et al., 1980a,b; Ipsen et al., 1990]. However, it has been noted that these parameters do not always yield a main transition at the expected temperature [Corvera et al.,1993], and so we will also propose our own values later on. The (dimensionless) variables Iα measure nematic chain order, and can be expressed in terms of the cross-sectional areas [Caill´e et al.,1980;Corvera et al.,1993]:
Iα=ωα
The last term in the Hamiltonian accounts for the interaction between the hydrophilic polar head groups and between them and water and also steric interactions from both head groups and the lipid chains. Although it is possible to consider a more realistic pairwise interaction between the head groups [Mouritsen,1984], this interaction can be approximated with a simple pressure term HP = ΠA, where Π is an effective lateral pressure acting on the lipid chains in the bilayer membrane, and A the total cross-sectional area occupied by the lipids chains,
A=
To study the phase behavior of the Pink model, we use the Monte Carlo (MC) simulation method. We mostly use triangular lattices of size N = L×L with periodic boundary